Fluctuation-dissipation theorem for thermo-refractive noise

We introduce a simple prescription for calculating the spectra of thermal fluctuations of temperature-dependent quantities of the form I over(T, I?) (t) = a?<d3 over(r, a??) I T (over(r, a??), t) q (over(r, a??)). Here T (over(r, a??), t) is the local temperature at location over(r, a??) and time t, and q (over(r, a??)) is an arbitrary function. As an example of a possible application, we compute the spectrum of thermo-refractive coating noise in LIGO, and find a complete agreement with the previous calculation of Braginsky, Gorodetsky and Vyatchanin. Our method has computational advantage, especially for non-regular or non-symmetric geometries, and for the cases where q (over(r, a??)) is non-negligible in a significant fraction of the total volume.